## Fractions: 1/5 = 1/10 = 1/80 = 1?

[Feature photo is a screen shot from the video “the sausages sharing episode,” see below.]

How in the world can 1/5 be the same as 1/10? Or 1/80 be the same as one whole thing? Such nonsense!

No, not nonsense. This is real-world common sense from a couple of boys faced with a problem just inside the edge of their ability — a problem that stretches them, but that they successfully solve, with a bit of gentle help on vocabulary.

Here’s the problem:

• How can you divide eight sausages evenly among five people?

Think for a moment about how you (or your child) might solve this puzzle, and then watch the video below.

## Egyptian Math: Fractions

I have been enjoying James Tanton’s website. In this video, Tanton explains a foolproof method for creating Egyptian fractions:

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## Egyptian Fractions: The Answer Sheet

Remember the Math Adventurer’s Rule: Figure it out for yourself! Whenever I give a problem in an Alexandria Jones story, I will try to post the answer (relatively) soon afterward. But don’t peek! If I tell you the answer, you miss out on the fun of solving the puzzle. So if you haven’t worked these problems yet, go back to the original post. Figure them out for yourself — and then check the answers just to prove that you got them right.

## The Secret of Egyptian Fractions

Alex made a poster of Egyptian-style fractions, from 1/2 to 9/10. Many of the fractions were easy. She knew that…

$\frac{5}{10} = \frac{4}{8} = \frac{3}{6} = \frac{2}{4} = \frac{1}{2}$

Therefore, as soon as she figured out one fraction, she had the answer to all of its equivalents.

She had the most trouble with the 7ths and 9ths. She tried converting these to other fractions that were easier to work with. For example, 28 has more factors than 7, making 28ths easier to break up into other fractions with one in the numerator.

## The Secret of Egyptian Fractions

Photo from Library of Congress via pingnews.

Archaeology professor Dr. Fibonacci Jones came home from a long day of lecturing and office work. Stepping inside the front door, he held up a shiny silver disk.

“Ta-da!” he said.

“All right!” said his daughter Alexandria. “The photos are here.”

They had to chase Alex’s brother Leon off the computer so they could view the images on the CD, but that wasn’t hard. He wanted to see the artifacts, too. Alex recognized several of the items they had dug up from the Egyptian scribe’s burial plot: the wooden palette, some clay pots, and of course the embalmed body.

Then came several close-up pictures of writing on papyrus.

Photo from MathsNet.net.

## How to Write Egyptian Fractions

“I remember how to read the Egyptian numbers,” Alex said, “but what are these marks above them?”

Dr. Jones nodded. “I thought you would catch that. Those are fractions. The scribe places an open mouth, which is the hieroglyph ‘r’, over a number to make its reciprocal.”

“I know that word,” Leon said. “It means one over the number. Like, the reciprocal of 12 is 1/12, right?”

“That is right. 1/12 would be written as…”

## The Rest of the Story

As I transcribed this article from my old math newsletter, I realized that it would require more graphics than I was willing to construct. LaTex does not handle Egyptian hieroglyphs — or at least, I don’t know how to make it do so. Instead, I decided to scan the newsletter pages and give them to you as a pdf file:

The file includes a student worksheet for Egyptian fractions from 1/2 to 9/10.